Systems and methods for optimization of time evolution for quantum computer-based eigenvalue estimation

ABSTRACT

A method may include: a computer program populating a Hermitian matrix A with input data; calculating an upper bound a for a maximum eigenvalue for the Hermitian matrix A; initializing a time evolution value t=1/a; generating a first quantum computer program using the time evolution value t; communicating the first quantum computer program to a quantum computer; receiving a result including a binary value for each n-bit string and a probability for each binary value; converting each binary value into an integer; identifying a maximum absolute value of the integers; determining a value x for the maximum absolute value of all of the integers; updating the time evolution value t based on the value of x; generating a second quantum computer program using the updated time evolution value t; and communicating, by the classical computer program, the second quantum computer program to the quantum computer.

BACKGROUND OF THE INVENTION 1. Field of the Invention

Embodiments are generally directed to systems and methods foroptimization of time evolution for quantum computer-based eigenvalueestimation.

2. Description of the Related Art

Time evolution is used in many calculations. A conventional timeevolution procedure works as follows: given a Hermitian matrix A and atime evolution (denoted t), the unitary U is built as U=e^(2πiAt). Fromthis, the eigenvalues for U may be estimated. Due to the 2π-periodicityof the imaginary exponential, the estimation of the eigenvalues is basedon the fractional part of λt, with λ an eigenvalue of A. If t isover-estimated (λt>½), information is lost and the result is false.Finally, an estimation method is used to obtain the first n-bits in thebinary decomposition of λt (i.e. 2nλt). If t is underestimated, thisleads to a sub-optimal precision as we will be using less than n bits torepresent the eigenvalues of A.

SUMMARY OF THE INVENTION

Systems and methods for optimization of time evolution for quantumcomputer-based eigenvalue estimation are disclosed. In one embodiment, amethod for a method for optimization of time evolution of quantumcomputer-based eigenvalue estimation may include: (1) receiving, by aclassical computer program executed by a classical computer, input data;(2) populating, by the classical computer program, a Hermitian matrix Awith the input data; (3) calculating, by the classical computer program,an upper bound a for a maximum eigenvalue (in modulo) for the Hermitianmatrix A; (4) initializing, by the classical computer program, a timeevolution value t, wherein t=1/a; (5) generating, by the classicalcomputer program, a first quantum computer program using the timeevolution value t; (6) communicating, by the classical computer program,the first quantum computer program to a quantum computer, wherein thequantum computer may be configured to execute the first quantum computerprogram; (7) receiving, by the classical computer program, a result ofthe execution of the first quantum computer program, wherein the resultmay include a binary value for each n-bit string and a probability foreach binary value; (8) converting, by the classical computer program,each binary value into an integer; (9) identifying, by the classicalcomputer program, a maximum absolute value of the integers; (10)determining, by the classical computer program, a value x for themaximum absolute value of all of the integers; (11) updating, by theclassical computer program, the time evolution value t based on thevalue of x; (12) generating, by the classical computer program, a secondquantum computer program using the updated time evolution value t; and(13) communicating, by the classical computer program, the secondquantum computer program to a quantum computer, wherein the quantumcomputer may be configured to execute the second quantum computerprogram.

In one embodiment, the input data may include market data, productiondata, or scheduling data.

In one embodiment, the upper bound a is equal to 2*sqrt(tr(A*A)), wheresqrt is the square root function, tr is a trace operator and A* is theconjugate transpose of the Hermitian matrix A.

In one embodiment, the method may further include filtering, by theclassical computer program, n-bit strings having probabilities below anoise level.

In one embodiment, the noise level may be based on a number of gates inthe quantum computer program and an infidelity level of gates in thequantum computer.

In one embodiment, the second quantum computer program may include aquantum Hamiltonian evolution circuit.

In one embodiment, the time evolution value t is updated when2^(n−1)−1−x is less than or equal to 1.

In one embodiment, the step of updating the time evolution value t basedon the maximum value of x may include: setting the time evolution valuet to t=t*2^(n) in response to the value of x being zero; or setting thetime evolution value t to t=t*2^(n−1)/x in response to the value of xnot being equal to zero.

In another embodiment, an electronic device may include: a memorystoring a classical computer program and a computer processor. Theclassical computer program is configured to: receive input data;populate a Hermitian matrix A with the input data; calculate an upperbound a for a maximum eigenvalue (in modulo) for the Hermitian matrix A;initialize a time evolution value t, wherein t=1/a; generate a firstquantum computer program using the time evolution value t; communicatethe first quantum computer program to a quantum computer, wherein thequantum computer is configured to execute the first quantum computerprogram; receive a result of the execution of the first quantum computerprogram, wherein the result may include a binary value for each n-bitstring and a probability for each binary value; convert each binaryvalue into an integer; identify a maximum absolute value of theintegers; determine a value x for the maximum absolute value of all ofthe integers; update the time evolution value t based on the value of x;generate a second quantum computer program using the updated timeevolution value t; and communicate the second quantum computer programto a quantum computer, wherein the quantum computer is configured toexecute the second quantum computer program.

In one embodiment, the input data may include market data, productiondata, or scheduling data.

In one embodiment, the upper bound a is equal to 2*sqrt(tr(A*A)), wheresqrt is the square root function, tr is a trace operator and A* is theconjugate transpose of the Hermitian matrix A.

In one embodiment, the classical computer program may be furtherconfigured to filter n-bit strings having probabilities below a noiselevel.

In one embodiment, the noise level may be based on a number of gates inthe quantum computer program and an infidelity level of gates in thequantum computer.

In one embodiment, the second quantum computer program may include aquantum Hamiltonian evolution circuit.

In one embodiment, the time evolution value t is updated when2^(n−1)−1−x is less than or equal to 1.

In one embodiment, the classical computer program may be configured toset the time evolution value t to t=t*2^(n) in response to the value ofx being zero; or setting the time evolution value t to t=t*2^(n−1)/x inresponse to the value of x not being equal to zero.

In another embodiment, a system may include: an electronic devicecomprising a memory storing a classical computer program and a computerprocessor; and a quantum computer in communication with the electronicdevice. The classical computer program is configured to receive inputdata, to populate a Hermitian matrix A with the input data, calculate anupper bound a for a maximum eigenvalue (in modulo) for the Hermitianmatrix A, initialize a time evolution value t, wherein t=1/a, generate afirst quantum computer program using the time evolution value t, andcommunicate the first quantum computer program to a quantum computer.The quantum computer is configured to execute the first quantum computerprogram. The classical computer program is configured to receive aresult of the execution of the first quantum computer program, whereinthe result may include a binary value for each n-bit string and aprobability for each binary value, convert each binary value into aninteger, identify a maximum absolute value of the integers, determine avalue x for the maximum absolute value of all of the integers, updatethe time evolution value t based on the value of x, generate a secondquantum computer program using the updated time evolution value t, andcommunicate the second quantum computer program to a quantum computer.The quantum computer is configured to execute the second quantumcomputer program.

In one embodiment, the upper bound a is equal to 2*sqrt(tr(A*A)), wheresqrt is the square root function, tr is a trace operator and A* is theconjugate transpose of the Hermitian matrix A.

In one embodiment, the classical computer program is further configuredto filter n-bit strings having probabilities below a noise level,wherein the noise level is based on a number of gates in the quantumcomputer program and an infidelity level of gates in the quantumcomputer.

In one embodiment, the time evolution value t is updated when2^(n−1)−1−x is less than or equal to 1, the time evolution value is setto t to t=t*2^(n) in response to the value of x being zero or tot=t*2^(n−1)/x in response to the value of x not being equal to zero.

BRIEF DESCRIPTION OF THE DRAWINGS

In order to facilitate a fuller understanding of the present invention,reference is now made to the attached drawings. The drawings should notbe construed as limiting the present invention but are intended only toillustrate different aspects and embodiments.

FIG. 1 depicts an exemplary system for optimization of time evolutionfor quantum computer-based eigenvalue estimation according to anembodiment;

FIG. 2 depicts a method for optimization of time evolution for quantumcomputer-based eigenvalue estimation according to an embodiment.

DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS

Systems and methods for optimization of time evolution for quantumcomputer-based eigenvalue estimation are disclosed.

Referring to FIG. 1 , a system for optimization of time evolution forquantum computer-based eigenvalue estimation is disclosed according toan embodiment. System 100 may include quantum computer 110 that mayexecute quantum computer program 115. Classical computer 120 mayinterface with quantum computer program 115 using computer program 125.Classical computer 120 may be any suitable classical computing device,including servers, workstations, desktop, notebook, laptop, or tabletcomputers, etc.

Classical computer program 125 may provide input to, and receive outputfrom, quantum computer 110 and/or quantum computer program 115. In oneembodiment, classical computer program 125 may provide quantum computerprogram 115 to quantum computer 110. Classical computer program 125 maydisplay the results of the execution of quantum computer program 115 ona display.

Database 130 may be a source of data that may be used to determine thetime series value, such as production data (e.g., cost of raw materials,number of widgets produced by a machine per hour, consumption of thismachine, etc.) for production optimization, market data (e.g., holdings,asset costs, etc.) for portfolio optimization, scheduling information(e.g., time per task, number of assets to perform the tasks,sequentiality of the tasks, etc.) for scheduling optimization, etc.

In one embodiment, classical computer program 125 may calculate a meanout of time series for the data from database 130 and may provide thetime series value to quantum computer program 115. Classical computerprogram 125 may further generate quantum computer program 115 (e.g., aHamiltonian evolution circuit) to be executed by quantum computer 110.Quantum computer 110 may then return results to classical computerprogram 125 for additional processing.

Referring to FIG. 2 , a method for optimization of time evolution forquantum computer-based eigenvalue estimation is disclosed according toan embodiment. In one embodiment, the method may be performed by acomputer program executed by a computing device in conjunction with aquantum computer program executed by a quantum computer or quantumcomputer simulator.

In step 200, input data for a calculation of interest may be receivedfrom a database. For example, the input data may be historical marketdata, production data, scheduling data, etc. Any suitable input data forwhich optimization may be desired may be used as is necessary and/ordesired.

In step 205, a matrix A, which may be a Hermitian matrix, may bepopulated with the input data.

In step 210, an upper bound value a may be calculated for lmax, themaximum eigenvalue (in modulo) for matrix A. For example, the upperbound a may be set equal to 2*sqrt(tr(A*A)), where sqrt is the squareroot function, tr is a trace operator and A* is the conjugate transposeof the Hermitian matrix A.

In step 215, the time evolution value t may be initialized using theequation t=1/a.

In step 220, the classical computer program may generate a quantumcomputer program using the value t and may send the quantum computerprogram to a quantum computer for execution. For example, the classicalcomputer program may convert the unitary U=e^(2πiAt) into a quantumcomputer program. This unitary depends on t and will be transformed intogates for the quantum computer. Thus, the value t directly impacts thegates. In embodiments, unitary decomposition may be used to decomposethe unitary into a quantum program subcomponent. This subcomponent maythen be used in a larger quantum computer program that may solve alinear system (e.g., the optimization problem). An example of such isthe HHL algorithm described in Harrow et al., “Quantum algorithm forsolving linear systems of equations,” available athttps://arxiv.org/abs/0811.3171 and Rebentrost et al., “Quantumcomputational finance: quantum algorithm for portfolio optimization,”available at https://arxiv.org/pdf/1811.03975.pdf, the disclosures ofwhich are hereby incorporated, by reference, in their entireties.

In step 225, the quantum computer may execute the quantum computerprogram and may return the results to the classical computer program.The results may be provided as a probability for each n-bit stringpossible, such as binary states with a probability linked to each state.An extract of the results for n=4 is provided in Table I below:

TABLE I Binary Probability 1110 12% 1111 13% 0000 21% 0001 11% 0011  9%0100 10%

In step 230, the classical computer program may extract a value x fromthe results. The classical computer program, knowing the input t, mayconvert the filtered results into meaningful values: Value x in TABLEIII. x_(i) may represent the binary output of the quantum computerresults, and x_(i)/(2^(n)*t) is the Value x.

In one embodiment, to evaluate positive and negatives values, the two'scomplement encoding method may be used. The first bit corresponds to thesign of the bit. An example integer and value for each binary isprovided in Table II below:

TABLE II Binary Probability Integer 1110 12% −2 1111 13% −1 0000 21% 00001 11% 1 0011  9% 3 0100 10% 4

Next, once the integer is known, the value of x may be determined bydividing the integer by t*2^(n). Using t=50 and n=4, the x values foreach integer is illustrated in Table III below:

TABLE III Binary Probability Integer Value x 1110 12% −2 −0.0025 111113% −1 −0.00125 0000 21% 0 0 0001 11% 1 0.00125 0011  9% 3 0.00375 010010% 4 0.0050

Next, the maximum absolute value of all integers is selected. In thiscase, the maximum absolute value of all integers is 4.

In one embodiment, rather than calculate a value x for each binary, themaximum absolute value may of all integers may be determined and thenthe value x may be calculated. These values, however, may provideestimated eigenvalues for other uses, comparisons, testing, etc.

To account for noise, the infidelity of each gate in quantum computer Eis determined, such as from a specification sheet for the quantumcomputer. Using the number of gates N in the quantum computer program,the classical computer program may then discard all states in the resultthat has a probability bellow E*N. In one embodiment, the filtering mayoccur before or after the conversion.

In step 235, if the value 2^(n−1)−1−x is less than or equal to one, theprocess continues to step 240, the classical computer program maygenerate a new quantum computer program for the value t and may send thenew quantum computer program to the quantum computer for execution. Inthe example above, the value 2^(n−1)−1−x is 3, which is greater thanone.

In step 245, if x equals 0, in step 250, the value t is set tot=t*2^(n). If the value of x is not equal to 0, in step 255, the value tis set to t=t*2^((n−1))/x. In both cases, the process continues withstep 220. In the example, x is not 0, so t is set to t*2^(n+1)/x, or100. The process continues using the value t=100.

Although several embodiments have been disclosed, it should berecognized that these embodiments are not exclusive to each other, andcertain elements or features from one embodiment may be used withanother.

Hereinafter, general aspects of implementation of the systems andmethods of the invention will be described.

The system of the invention or portions of the system of the inventionmay be in the form of a “processing machine,” such as a general-purposecomputer, for example. As used herein, the term “processing machine” isto be understood to include at least one processor that uses at leastone memory. The at least one memory stores a set of instructions. Theinstructions may be either permanently or temporarily stored in thememory or memories of the processing machine. The processor executes theinstructions that are stored in the memory or memories in order toprocess data. The set of instructions may include various instructionsthat perform a particular task or tasks, such as those tasks describedabove. Such a set of instructions for performing a particular task maybe characterized as a program, software program, or simply software.

In one embodiment, the processing machine may be a specializedprocessor.

As noted above, the processing machine executes the instructions thatare stored in the memory or memories to process data. This processing ofdata may be in response to commands by a user or users of the processingmachine, in response to previous processing, in response to a request byanother processing machine and/or any other input, for example.

As noted above, the processing machine used to implement the inventionmay be a general-purpose computer. However, the processing machinedescribed above may also utilize any of a wide variety of othertechnologies including a special purpose computer, a computer systemincluding, for example, a microcomputer, mini-computer or mainframe, aprogrammed microprocessor, a micro-controller, a peripheral integratedcircuit element, a CSIC (Customer Specific Integrated Circuit) or ASIC(Application Specific Integrated Circuit) or other integrated circuit, alogic circuit, a digital signal processor, a programmable logic devicesuch as a FPGA, PLD, PLA or PAL, or any other device or arrangement ofdevices that is capable of implementing the steps of the processes ofthe invention.

In one embodiment, the processing machine may be a classical computer, aquantum computer, etc.

It is appreciated that in order to practice the method of the inventionas described above, it is not necessary that the processors and/or thememories of the processing machine be physically located in the samegeographical place. That is, each of the processors and the memoriesused by the processing machine may be located in geographically distinctlocations and connected so as to communicate in any suitable manner.Additionally, it is appreciated that each of the processor and/or thememory may be composed of different physical pieces of equipment.Accordingly, it is not necessary that the processor be one single pieceof equipment in one location and that the memory be another single pieceof equipment in another location. That is, it is contemplated that theprocessor may be two pieces of equipment in two different physicallocations. The two distinct pieces of equipment may be connected in anysuitable manner. Additionally, the memory may include two or moreportions of memory in two or more physical locations.

To explain further, processing, as described above, is performed byvarious components and various memories. However, it is appreciated thatthe processing performed by two distinct components as described abovemay, in accordance with a further embodiment of the invention, beperformed by a single component. Further, the processing performed byone distinct component as described above may be performed by twodistinct components. In a similar manner, the memory storage performedby two distinct memory portions as described above may, in accordancewith a further embodiment of the invention, be performed by a singlememory portion. Further, the memory storage performed by one distinctmemory portion as described above may be performed by two memoryportions.

Further, various technologies may be used to provide communicationbetween the various processors and/or memories, as well as to allow theprocessors and/or the memories of the invention to communicate with anyother entity; i.e., so as to obtain further instructions or to accessand use remote memory stores, for example. Such technologies used toprovide such communication might include a network, the Internet,Intranet, Extranet, LAN, an Ethernet, wireless communication via celltower or satellite, or any client server system that providescommunication, for example. Such communications technologies may use anysuitable protocol such as TCP/IP, UDP, or OSI, for example.

As described above, a set of instructions may be used in the processingof the invention. The set of instructions may be in the form of aprogram or software. The software may be in the form of system softwareor application software, for example. The software might also be in theform of a collection of separate programs, a program module within alarger program, or a portion of a program module, for example. Thesoftware used might also include modular programming in the form ofobject-oriented programming. The software tells the processing machinewhat to do with the data being processed.

Further, it is appreciated that the instructions or set of instructionsused in the implementation and operation of the invention may be in asuitable form such that the processing machine may read theinstructions. For example, the instructions that form a program may bein the form of a suitable programming language, which is converted tomachine language or object code to allow the processor or processors toread the instructions. That is, written lines of programming code orsource code, in a particular programming language, are converted tomachine language using a compiler, assembler or interpreter. The machinelanguage is binary coded machine instructions that are specific to aparticular type of processing machine, i.e., to a particular type ofcomputer, for example. The computer understands the machine language.

Also, the instructions and/or data used in the practice of the inventionmay utilize any compression or encryption technique or algorithm, as maybe desired. An encryption module might be used to encrypt data. Further,files or other data may be decrypted using a suitable decryption module,for example.

As described above, the invention may illustratively be embodied in theform of a processing machine, including a computer or computer system,for example, that includes at least one memory. It is to be appreciatedthat the set of instructions, i.e., the software for example, thatenables the computer operating system to perform the operationsdescribed above may be contained on any of a wide variety of media ormedium, as desired. Further, the data that is processed by the set ofinstructions might also be contained on any of a wide variety of mediaor medium. That is, the particular medium, i.e., the memory in theprocessing machine, utilized to hold the set of instructions and/or thedata used in the invention may take on any of a variety of physicalforms or transmissions, for example. Illustratively, the medium may bein the form of paper, paper transparencies, a compact disk, a DVD, anintegrated circuit, a hard disk, a floppy disk, an optical disk, amagnetic tape, a RAM, a ROM, a PROM, an EPROM, a wire, a cable, a fiber,a communications channel, a satellite transmission, a memory card, a SIMcard, a memory stick, or other remote transmission, as well as any othermedium or source of data that may be read by the processors of theinvention.

Further, the memory or memories used in the processing machine thatimplements the invention may be in any of a wide variety of forms toallow the memory to hold instructions, data, or other information, as isdesired. Thus, the memory might be in the form of a database to holddata. The database might use any desired arrangement of files such as aflat file arrangement or a relational database arrangement, for example.

In the system and method of the invention, a variety of “userinterfaces” may be utilized to allow a user to interface with theprocessing machine or machines that are used to implement the invention.As used herein, a user interface includes any hardware, software, orcombination of hardware and software used by the processing machine thatallows a user to interact with the processing machine. A user interfacemay be in the form of a dialogue screen for example. A user interfacemay also include any of a mouse, touch screen, keyboard, keypad, voicereader, voice recognizer, dialogue screen, menu box, list, checkbox,toggle switch, a pushbutton or any other device that allows a user toreceive information regarding the operation of the processing machine asit processes a set of instructions and/or provides the processingmachine with information. Accordingly, the user interface is any devicethat provides communication between a user and a processing machine. Theinformation provided by the user to the processing machine through theuser interface may be in the form of a command, a selection of data, orsome other input, for example.

As discussed above, a user interface is utilized by the processingmachine that performs a set of instructions such that the processingmachine processes data for a user. The user interface is typically usedby the processing machine for interacting with a user either to conveyinformation or receive information from the user. However, it should beappreciated that in accordance with some embodiments of the system andmethod of the invention, it is not necessary that a human user actuallyinteract with a user interface used by the processing machine of theinvention. Rather, it is also contemplated that the user interface ofthe invention might interact, i.e., convey and receive information, withanother processing machine, rather than a human user. Accordingly, theother processing machine might be characterized as a user. Further, itis contemplated that a user interface utilized in the system and methodof the invention may interact partially with another processing machineor processing machines, while also interacting partially with a humanuser.

It will be readily understood by those persons skilled in the art thatthe present invention is susceptible to broad utility and application.Many embodiments and adaptations of the present invention other thanthose herein described, as well as many variations, modifications andequivalent arrangements, will be apparent from or reasonably suggestedby the present invention and foregoing description thereof, withoutdeparting from the substance or scope of the invention.

Accordingly, while the present invention has been described here indetail in relation to its exemplary embodiments, it is to be understoodthat this disclosure is only illustrative and exemplary of the presentinvention and is made to provide an enabling disclosure of theinvention. Accordingly, the foregoing disclosure is not intended to beconstrued or to limit the present invention or otherwise to exclude anyother such embodiments, adaptations, variations, modifications orequivalent arrangements.

What is claimed is:
 1. A method for a method for optimization of timeevolution of quantum computer-based eigenvalue estimation comprising:receiving, by a classical computer program executed by a classicalcomputer, input data; populating, by the classical computer program, aHermitian matrix A with the input data; calculating, by the classicalcomputer program, an upper bound a for a maximum eigenvalue (in modulo)for the Hermitian matrix A; initializing, by the classical computerprogram, a time evolution value t, wherein t=1/a; generating, by theclassical computer program, a first quantum computer program using thetime evolution value t; communicating, by the classical computerprogram, the first quantum computer program to a quantum computer,wherein the quantum computer is configured to execute the first quantumcomputer program; receiving, by the classical computer program, a resultof the execution of the first quantum computer program, wherein theresult comprises a binary value for each n-bit string and a probabilityfor each binary value; converting, by the classical computer program,each binary value into an integer; identifying, by the classicalcomputer program, a maximum absolute value of the integers; determining,by the classical computer program, a value x for the maximum absolutevalue of all of the integers; updating, by the classical computerprogram, the time evolution value t based on the value of x; generating,by the classical computer program, a second quantum computer programusing the updated time evolution value t; and communicating, by theclassical computer program, the second quantum computer program to thequantum computer, wherein the quantum computer is configured to executethe second quantum computer program.
 2. The method of claim 1, whereinthe input data comprises market data, production data, or schedulingdata.
 3. The method of claim 1, wherein the upper bound a is equal to2*sqrt(tr(A*A)), where sqrt is the square root function, tr is a traceoperator and A* is the conjugate transpose of the Hermitian matrix A. 4.The method of claim 1, further comprising: filtering, by the classicalcomputer program, n-bit strings having probabilities below a noiselevel.
 5. The method of claim 4, wherein the noise level is based on anumber of gates in the quantum computer program and an infidelity levelof gates in the quantum computer.
 6. The method of claim 1, wherein thesecond quantum computer program comprises a quantum Hamiltonianevolution circuit.
 7. The method of claim 1, wherein the time evolutionvalue t is updated when 2^(n−1)−1−x is less than or equal to
 1. 8. Themethod of claim 1, wherein the step of updating the time evolution valuet based on the maximum value of x comprises: setting the time evolutionvalue t to t=t*2^(n) in response to the value of x being zero; orsetting the time evolution value t to t=t*2^(n−1)/x in response to thevalue of x not being equal to zero.
 9. An electronic device comprising:a memory storing a classical computer program; and a computer processor;wherein the classical computer program is configured to: receive inputdata; populate a Hermitian matrix A with the input data; calculate anupper bound a for a maximum eigenvalue (in modulo) for the Hermitianmatrix A; initialize a time evolution value t, wherein t=1/a; generate afirst quantum computer program using the time evolution value t;communicate the first quantum computer program to a quantum computer,wherein the quantum computer is configured to execute the first quantumcomputer program; receive a result of the execution of the first quantumcomputer program, wherein the result comprises a binary value for eachn-bit string and a probability for each binary value; convert eachbinary value into an integer; identify a maximum absolute value of theintegers; determine a value x for the maximum absolute value of all ofthe integers; update the time evolution value t based on the value of x;generate a second quantum computer program using the updated timeevolution value t; and communicate the second quantum computer programto the quantum computer, wherein the quantum computer is configured toexecute the second quantum computer program.
 10. The electronic deviceof claim 9, wherein the input data comprises market data, productiondata, or scheduling data.
 11. The electronic device of claim 9, whereinthe upper bound a is equal to 2*sqrt(tr(A*A)), where sqrt is the squareroot function, tr is a trace operator and A* is the conjugate transposeof the Hermitian matrix A.
 12. The electronic device of claim 9, whereinthe classical computer program is further configured to filter n-bitstrings having probabilities below a noise level.
 13. The electronicdevice of claim 12, wherein the noise level is based on a number ofgates in the quantum computer program and an infidelity level of gatesin the quantum computer.
 14. The electronic device of claim 9, whereinthe second quantum computer program comprises a quantum Hamiltonianevolution circuit.
 15. The electronic device of claim 9, wherein thetime evolution value t is updated when 2^(n−1)−1−x is less than or equalto
 1. 16. The electronic device of claim 9, the classical computerprogram is configured to set the time evolution value t to t=t*2^(n) inresponse to the value of x being zero or to t=t*2^(n−1)/x in response tothe value of x not being equal to zero.
 17. A system, comprising: anelectronic device comprising a memory storing a classical computerprogram and a computer processor; and a quantum computer incommunication with the electronic device; wherein the classical computerprogram is configured to receive input data; the classical computerprogram is configured to populate a Hermitian matrix A with the inputdata; the classical computer program is configured to calculate an upperbound a for a maximum eigenvalue (in modulo) for the Hermitian matrix A;the classical computer program is configured to initialize a timeevolution value t, wherein t=1/a; the classical computer program isconfigured to generate a first quantum computer program using the timeevolution value t; the classical computer program is configured tocommunicate the first quantum computer program to a quantum computer;the quantum computer is configured to execute the first quantum computerprogram; the classical computer program is configured to receive aresult of the execution of the first quantum computer program, whereinthe result comprises a binary value for each n-bit string and aprobability for each binary value; the classical computer program isconfigured to convert each binary value into an integer; the classicalcomputer program is configured to identify a maximum absolute value ofthe integers; the classical computer program is configured to determinea value x for the maximum absolute value of all of the integers; theclassical computer program is configured to update the time evolutionvalue t based on the value of x; the classical computer program isconfigured to generate a second quantum computer program using theupdated time evolution value t; the classical computer program isconfigured to communicate the second quantum computer program to thequantum computer; and the quantum computer is configured to execute thesecond quantum computer program.
 18. The system of claim 17, wherein theupper bound a is equal to 2*sqrt(tr(A*A)), where sqrt is the square rootfunction, tr is a trace operator and A* is the conjugate transpose ofthe Hermitian matrix A.
 19. The system of claim 17, wherein theclassical computer program is further configured to filter n-bit stringshaving probabilities below a noise level, wherein the noise level isbased on a number of gates in the quantum computer program and aninfidelity level of gates in the quantum computer.
 20. The system ofclaim 17, wherein the time evolution value t is updated when 2^(n−1)−1−xis less than or equal to 1; and the time evolution value t is set tot=t*2^(n) in response to the value of x being zero or to t=t*2^(n−1)/xin response to the value of x not being equal to zero.